Programming Languages Featherweight Java David Walker

Transcription

1 Programming Languages Featherweight Java David Walker

2 Overview Featherweight Java (FJ), a minimal Javalike language. Models inheritance and subtyping. Immutable objects: no mutation of fields. Trivialized core language. 1

3 Abstract Syntax The abstract syntax of FJ is given by the following grammar: Classes C : : = class c extends c {c f; k d} Constructors k : : = c(c x) {super(x); this.f =x;} Methods d : : = c m(c x) {return e;} Types τ : : = c Expressions e : : = x e.f e.m(e) new c(e) (c) e Underlining indicates one or more. If e appears in an inference rule and e i does too, there is an implicit understandng that e i is one of the e s in e. And similarly with other underlined constructs. 2

4 Abstract Syntax Classes in FJ have the form: class c extends c {c f; k d} Class c is a sub-class of class c. Constructor k for instances of c. Fields c f. Methods d. 3

5 Abstract Syntax Constructor expressions have the form c(c x, c x) {super(x ); this.f=x;} Arguments correspond to super-class fields and sub-class fields. Initializes super-class. Initializes sub-class. 4

6 Abstract Syntax Methods have the form c m(c x) {return e;} Result class c. Argument class(es) c. Binds x and this in e. 5

7 Abstract Syntax Minimal set of expressions: Field selection: e.f. Message send: e.m(e). Instantiation: new c(e). Cast: (c) e. 6

8 FJ Example class Pt extends Object { int x; int y; } Pt (int x, int y) { super(); this.x = x; this.y = y; } int getx () { return this.x; } int gety () { return this.y; } 7

9 FJ Example class CPt extends Pt { color c; } CPt (int x, int y, color c) { super(x,y); this.c = c; } color getc () { return this.c; } 8

10 Class Tables and Programs A class table T is a finite function assigning classes to class names. A program is a pair (T, e) consisting of A class table T. An expression e. 9

11 Static Semantics Judgement forms: τ <: τ c c Γ e : τ d ok in c c ok T ok fields(c) = c f type(m, c) = c c subtyping subclassing expression typing well-formed method well-formed class well-formed class table field lookup method type 10

12 Static Semantics Variables: Γ(x) = τ Γ x : τ Must be declared, as usual. Introduced within method bodies. 11

13 Static Semantics Field selection: Γ e 0 : c 0 fields(c 0 ) = c f Γ e 0.f i : c i Field must be present. Type is specified in the class. 12

14 Static Semantics Message send: Γ e 0 : c 0 Γ e : c type(m, c 0 ) = c c c <: c Γ e 0.m(e) : c Method must be present. Argument types must be subtypes of parameters. 13

15 Static Semantics Instantiation: Γ e : c c <: c fields(c) = c f Γ new c(e) : c Initializers must have subtypes of fields. 14

16 Static Semantics Casting: Γ e 0 : d Γ (c) e 0 : c All casts are statically acceptable! Could try to detect casts that are guaranteed to fail at run-time. 15

17 Subclassing Sub-class relation is implicitly relative to a class table. T (c) = class c extends c {...;......} c c Reflexivity, transitivity of sub-classing: (T (c) defined) c c c c c c c c Sub-classing only by explicit declaration! 16

18 Subtyping Subtyping relation: τ <: τ. τ <: τ τ <: τ τ <: τ τ <: τ c c c <: c Subtyping is determined solely by subclassing. 17

19 Class Formation Well-formed classes: k = c(c x, c x) {super(x ); this.f=x;} fields(c ) = c f d i ok in c class c extends c {c f; k d} ok Constructor has arguments for each superand sub-class field. Constructor initializes super-class before subclass. Sub-class methods must be well-formed relative to the super-class. 18

20 Class Formation Method overriding, relative to a class: T (c) = class c extends c {...;......} type(m, c ) = c c 0 x : c, this:c e 0 : c 0 c 0 <: c 0 c 0 m(c x) {return e 0 ;} ok in c Sub-class method must return a subtype of the super-class method s result type. Argument types of the sub-class method must be exactly the same as those for the super-class. Need another case to cover method extension. 19

21 Program Formation A class table is well-formed iff all of its classes are well-formed: c dom(t ) T (c) ok T ok A program is well-formed iff its class table is well-formed and the expression is well-formed: T ok e : τ (T, e) ok 20

22 Method Typing The type of a method is defined as follows: T (c) = class c extends c {...;... d} d i = c i m(c i x) {return e;} type(m i, c) = c i c i T (c) = class c extends c {...;... d} m / d type(m i, c ) = c i c i type(m, c) = c i c i 21

23 Dynamic Semantics Transitions: e T e. Transitions are indexed by a (well-formed) class table! Dynamic dispatch. Downcasting. We omit explicit mention of T in what follows. 22

24 Dynamic Semantics Object values have the form where new c(e, e) e are the values of the super-class fields. and e are the values of the sub-class fields. c indicates the true (dynamic) class of the instance. Use this judgement to affirm an expression is a value: new c(e, e) value Rules new Object value e i value e i value new c(e, e) value 23

25 Dynamic Semantics Field selection: fields(c) = c f, c f e value e value new c(e, e).f i e i fields(c) = c f, c f e value e value new c(e, e).f i e i Fields in sub-class must be disjoint from those in super-class. Selects appropriate field based on name. 24

26 Dynamic Semantics Message send: body(m, c) = x e 0 e value e value new c(e).m(e ) {e /x}{new c(e)/this}e 0 The identifier this stands for the object itself. Compare with recursive functions in MinML. 25

27 Dynamic Semantics Cast: c c e value (c ) new c(e) new c(e) No transition (stuck) if c is not a sub-class of c! Sh/could introduce error transitions for cast failure. 26

28 Dynamic Semantics Search rules (CBV): e 0 e 0 e 0.f e 0.f e 0 e 0 e 0.m(e) e 0.m(e) e 0 value e e e 0.m(e) e 0.m(e ) 27

29 Dynamic Semantics Search rules (CBV), cont d: e e new c(e) new c(e ) e 0 e 0 (c) e 0 (c) e 0 28

30 Dynamic Semantics Dynamic dispatch: T (c) = class c extends c {...;... d} d i = c i m(c i x) {return e;} body(m i, c) = x e T (c) = class c extends c {...;... d} m / d body(m, c ) = x e body(m, c) = x e Climbs the class hierarchy searching for the method. Static semantics ensures that the method must exist! 29

31 Type Safety Theorem 1 (Preservation) Assume that T is a well-formed class table. If e : τ and e e, then e : τ for some τ <: τ. Proved by induction on transition relation. Type may get smaller during execution due to casting! 30

32 Type Safety Lemma 2 (Canonical Forms) If e : c and e value, then e = new d(e 0 ) with d c and e 0 value. Values of class type are objects (instances). The dynamic class of an object may be lower in the subtype hierarchy than the static class. 31

33 Type Safety Theorem 3 (Progress) Assume that T is a well-formed class table. If e : τ then either 1. v value, or 2. e has the form (c) new d(e 0 ) with e 0 value and d c, or 3. there exists e such that e e. 32

34 Type Safety Comments on the progress theorem: Well-typed programs can get stuck! only because of a cast.... But Precludes message not understood error. Proof is by induction on typing. 33

35 Variations and Extensions A more flexible static semantics for override: Subclass result type is a subtype of the superclass result type. Subclass argument types are supertypes of the corresponding superclass argument types. 34

36 Variations and Extensions Java adds arrays and covariant array subtyping: τ <: τ τ [ ] <: τ [ ] What effect does this have? 35

37 Variations and Extensions Java adds array covariance: τ <: τ τ [ ] <: τ [ ] Perfectly OK for FJ, which does not support mutation and assignment. With assignment, might store a supertype value in an array of the subtype. Subsequent retrieval at subtype is unsound. Java inserts a per-assignment run-time check and exception raise to ensure safety. 36

38 Variations and Extensions Static fields: Must be initialized as part of the class definition (not by the constructor). In what order are initializers to be evaluated? Could require initialization to a constant. 37

39 Variations and Extensions Static methods: Essentially just recursive functions. No overriding. Static dispatch to the class, not the instance. 38

40 Variations and Extensions Final methods: Preclude override in a sub-class. Final fields: Sensible only in the presence of mutation! 39

41 Variations and Extensions Abstract methods: Some methods are undefined (but are declared). Cannot form an instance if any method is abstract. 40

42 Class Tables Type checking requires the entire program! Class table is a global property of the program and libraries. Cannot type check classes separately from one another. 41